PCA in Machine Learning & Data Science

Principal Component Analysis (PCA) in Data Science

PCA is a dimensionality reduction technique used to simplify complex datasets while preserving as much variability as possible. It does so by transforming the data into a new coordinate system defined by its principal components.

Key Concepts:

1. Eigenvectors and Eigenvalues:
2. Steps Involving Eigenvectors and Eigenvalues in PCA:

Why PCA and Eigen Concepts are Important in Data Science:

1. Dimensionality Reduction:
2. Feature Extraction:
3. Noise Reduction:
4. Data Visualization:
5. Preprocessing for Machine Learning Models:

Applications in Data Science:

• Image compression.
• Face recognition.
• Exploratory Data Analysis (EDA).
• Preprocessing high-dimensional data for classification or regression tasks.

Here’s a small Python example of PCA using NumPy:

import numpy as np

# Example dataset (3 samples, 2 features)
data = np.array([[2.5, 2.4],
                 [0.5, 0.7],
                 [2.2, 2.9]])

# Step 1: Standardize the data (mean = 0)
mean = np.mean(data, axis=0)
data_centered = data - mean

# Step 2: Compute the covariance matrix
cov_matrix = np.cov(data_centered.T)

# Step 3: Compute eigenvalues and eigenvectors
eigenvalues, eigenvectors = np.linalg.eig(cov_matrix)

# Step 4: Sort eigenvalues and eigenvectors
sorted_indices = np.argsort(eigenvalues)[::-1]
eigenvalues = eigenvalues[sorted_indices]
eigenvectors = eigenvectors[:, sorted_indices]

# Step 5: Transform data into the new PCA space
projected_data = np.dot(data_centered, eigenvectors)

# Output
print("Eigenvalues:", eigenvalues)
print("Eigenvectors:\n", eigenvectors)
print("Projected Data:\n", projected_data)
        

Explanation:

1. Data Standardization: Centers the data by subtracting the mean.
2. Covariance Matrix: Measures the relationships between features.
3. Eigenvalues and Eigenvectors: Determine the principal components.
4. Projection: Transforms the original data into the reduced space.